Rapid and stable determination of rotation matrices between spherical harmonics by direct recursion

Recurrence relations are derived for constructing rotation matrices between complex spherical harmonics directly as polynomials of the elements of the generating 3×3 rotation matrix, bypassing the intermediary of any parameters such as Euler angles. The connection to the rotation matrices for real spherical harmonics is made explicit. The recurrence formulas furnish a simple, efficient, and numerically stable evaluation procedure for the real and complex representations of the rotation group. The advantages over the Wigner formulas are documented. The results are relevant for directing atomic orbitals as well as multipoles.